The algorithm uses the L1c co-registered OLCI and SLSTR data product as input, projected on the OLCI grid. Co-registration is made based on the common 865 nm radiometric band. Over selected ground-control points, radiometric images of the SLSTR 865 nm band are extracted and compared to the OLCI 865 nm acquisitions. The OLCI image is moved around according to shift vectors and the cross-correlation with the fixed SLSTR window is calculated. The elements of the shift vectors at which a maximum in cross-correlation is reached determine the pixel deregistration between the OLCI and SLSTR reference channel.

Over ocean, AOD is returned using the full swath of the Level 1c (L1c) product (1400 km), while over land the region covered by both nadir and oblique view (750 km) is used for the best-quality retrieval, and aerosol retrieval is also made outside this region where both nadir-only SLSTR and OLCI are available (∼ 1200 km). Beginning with the L1c product, pixels are flagged to screen cloud, snow ice, or sunglint areas. In addition, all neighbouring pixels to cloud pixels are flagged to avoid edge effects. Pixels are grouped into “super-pixels” formed by blocks of 15 × 15 pixels of the L1c SYN pixels at 300 m spatial resolution. Thus, a super-pixel represents a resolution of about 4.5 km × 4.5 km. The result is a super-pixel giving aggregated cloud-free TOA radiance for the nadir and oblique view (if present) of the same surface location. The inversion is carried out for all land and ocean super-pixels which are at least 50 % free of cloud, ice, and snow. Over-ocean retrieval proceeds if either nadir or oblique super-pixels are valid, while over land both nadir and oblique must be valid for dual-view retrieval or nadir only for single-view (spectral) retrieval.

The basis of the algorithm is iterative non-linear optimisation to jointly retrieve aerosol optical depth at a reference wavelength of 550 nm, referred to as AOD550, and the fine-mode fraction (FMF) of AOD550. Atmospheric radiative transfer is approximated as a look-up table (LUT) to relate top-of-atmosphere to surface reflectance for a given estimate of aerosol parameters, water vapour, ozone, and surface pressure. Over both land and ocean, the retrieval requires optimisation of a cost function expressing the fit of derived surface reflectance to ocean or land models of reflectance. Several additional parameters are provided, which are derived from these properties, to provide information on spectral variation of AOD and surface reflectance values intended as diagnostics (see Sect. 2.3 for details). When a single viewing direction is used, the inversion is made over spectral bands in that direction only. This is normally the case outside the oblique view swath for which nadir only is used, but use of the oblique view alone also occurs over ocean where the nadir view is obscured by glint or cloud. Over ocean, only SLSTR channels (five spectral bands corresponding to S1 – 554 nm, S2 – 659 nm, S3 – 865 nm, S5 – 1613 nm, and S6 – 2255 nm) are taken into account in the aerosol retrieval. Over land, both sensors (including OLCI 442.5 nm spectral band) are considered.

A climatology of aerosol composition (Kinne et al., 2013; de Leeuw et al., 2015) is used to provide further information on the fine and coarse components (non-spherical vs. spherical, single-scattering albedo) and a prior estimate of the fine-mode fraction. We fit parameters for both AOD and FMF, which controls the spectral variation of AOD. Although AOD is parameterised by a single nominal wavelength (550 nm), all wavelengths of SLSTR, and additionally the 442.5 nm OLCI channel over land, are used in this fitting. The single-scattering albedo (SSA) is constrained by climatology for the coarse- and fine-mode extremes separately and as a priori information. The retrieval of FMF results in SSA by interpolation between these extremes; however, this should be seen as a potential diagnostic for retrieval performance rather than a user product. Further constraints prevent unfeasible retrieval (e.g. negative AOD or surface reflectance). An estimate of the 1 standard deviation (SD) error in AOD at 550 nm is derived from the second derivative (curvature) of the error surface near the optimal value.

Over ocean, a surface reflectance model gives a reflectance estimate determined from the wind speed and direction and using the models of Cox and Munk (1954) for glint, Monahan and O'Muircheartaigh (1980) and Koepke (1984) for foam fraction and spectral reflectance, and Morel's case I water reflectance model dependent on pigment concentration (Morel, 1988). The ocean inversion uses bands from SLSTR only, using both views to invert if both are available or a single view (either nadir or oblique) if one view is either obscured by cloud, contaminated by glint, or in a swath region where only a single view is present. For land, the reflectance constraint is the result of fitting to separate angular and spectral parameterised models (North, 2002; North et al., 2008; Davies and North, 2015; North and Heckel, 2019). When the oblique SLSTR view is not available, only the spectral constraint is used, allowing AOD estimation over the full L1c swath over both land and ocean.

A final step is used to filter residual cloud contamination or other sources of poor retrieval. This is based on thresholding of local image standard deviation, as discussed in Sogacheva et al. (2017). Over ocean, a final screening is also made on the quality of model fit. Any AOD value outside the AOD valid range of [0, 4] is replaced by a “fill” value of 6.53. A “clean-air” test is performed to recognise cases when an extensive rejection of low AOD values occurs in the case of a clean atmosphere, which often happens over dark surfaces. In the case that this test is positive, which is indicated by quality flags, a value of 0.04 is used.

During post-processing, further aerosol outputs are derived from the retrieved AOD550 and FM AOD. This includes spectral variation of AOD, which is given using a pre-computed look-up table from the retrieved FM AOD and aerosol mixture. The Angström exponent is computed based on a pair of spectral AOD values. Here we choose 865 and 550 nm. A full set of quality flags is provided.

Scatter density plots for S3A SY_2 AOD550 (syAOD550, or syAOD) and corresponding AERONET AOD550 (aAOD550, or aAOD) for all matchups available for the NH and SH, including binned AOD offsets, are shown in Fig. 2. For most of the matchups (91 %), syAOD is small (< 0.4).

Validation statistics for S3A and S3B products are shown in Table 1. These include the number of points (N), the percentage of matchups which fit into the MODIS AOD error envelope (EE) defined as ±0.05 ± 0.2 × AOD (Remer et al., 2013), the percentage of matchups which satisfy Global Climate Observing System (GCOS) requirements of 0.03 or 10 % of AOD (GCOS, 2016), the Pearson correlation coefficient (R), root mean square (rms), standard deviation (SD), and bias and slope defined with linear regression (polynomial fit) applied to all available matchups.

A difference in the algorithm performance in the NH and SH is clear. For S3A, the fraction of matchups in the EE (70.8 %) and the fraction of matchups which satisfy GCOS requirements (43.0 %) are considerably higher in the SH (in the NH, 48.2 % and 20.5 %, respectively), but R (0.62) and rms (0.22) are only slightly better (in the NH, 0.6 and 0.28, respectively). For all matchups, validation statistics are better for S3B: in the SH, more matchups fit the EE (74.6 %) and GCOS (44.9 %); R (0.70) is higher and rms (0.15) is lower. In the NH, the difference between S3A and S3B is smaller.

In addition to the statistics shown in Table 1, we performed a respective analysis for limited AOD ranges. For aAOD < 1.5, syAOD validation statistics are slightly better than statistics for all aAOD ranges: bias is close to 0.1, and slope is close to 1 for both S3A and S3B AOD products in the NH. For aAOD > 1.5, bias is ca. 1.3 in the NH (where N is 127 and 125 for S3A and S3B, respectively). In the SH matchups available for the S3B product are located close to the 1 : 1 line; however, the number of matchups with aAOD > 1.5 is too small (N is 3/2) to calculate validation statistics.

Group (dual, singleN, singleO) analysis reveals that most of the low-biased syAOD outliers were retrieved with the dual processor (Fig. 2), while most of the high-biased syAOD outliers were retrieved with the singleN processor. Total bias is smaller for the dual group globally and in both the NH and SH (Table 1). For aAOD < 1.5, syAOD bias is close to 0 for the dual group; for the singleN group bias is higher than for all matchups and increases with aAOD. Validation statistics are, in general, better in the SH (except for R for all the single groups). As for all matchups, validation statistics are slightly better for S3B.

Analysis of the binned (based on aAOD, bin size of 0.1) syAOD offsets to aAOD was carried out. For S3A (Fig. 3), the dual group shows better performance. In this group, the positive offset at low (< 0.2) AOD vanishes towards higher AOD and turns to negative at AOD > 0.4. About 91 % of matchups fit the AOD range of [0, 0.4]. In this AOD range, an offset is 0.03-0.05 higher in the NH compared with the SH. Offsets for the S3B in the same AOD range are lower (up to 0.03). Offsets for singleN and singleO groups are positive in the AOD range of [0, 1.2]. For high AOD, offsets are in general higher; however, less than 1.4 % of the matchups fit the range of aAOD > 1.

For the aAOD binned in 0.1 intervals, the global difference (dAOD) between syAOD and aAOD represented with the median bias and dAOD standard deviation is shown in Fig. 4 for all aerosol types including background (aAOD ≤ 0.2) AOD as well as fine-dominated and coarse-dominated AOD. Globally, background AOD (64 % from all matchups) is overestimated by 0.04–0.06. Overestimation of fine-dominated matchups increases from 0.07 to 0.15 in the AOD range of 0.2–1.2 (34 % of matchups). Overestimation for coarse-dominated matchups is about 0.05 for aAOD < 0.7; for aAOD of 0.7–0.9, an overestimation for coarse-dominated matchups is within the GCOS requirements of ± 0.03 dAOD. For aAOD > 1.2, dAOD varies in sign and amplitude; however, the number of matchups in this size range is low (< 1 %) and results are thus unstable. Fractions of the fine-dominated matchups per bin are 60 %–70 % for aAOD in the range of 0.2–0.9 and more than 70 % for aAOD > 0.9. Thus, binned offsets for all matchups closely follow offsets for fine-dominated matchups.

In the NH, the syAOD offset for the background matchups is ∼ 0.07; in the SH the offset is lower (< 0.02). Binned offsets for the fine-dominated and coarse-dominated matchups in the NH are similar to those for the globe. In the SH, offsets of syAOD are higher for aAOD > 0.4, where the number of the matchups per bin is low (< 50).

Monthly (January, February, March, etc.), seasonal (DJF, MAM, JJA, SON), and annual (year) variations of the validation results for S3A and S3B syAOD550 for the globe, NH, and SH are shown in Fig. 5.

The correlation coefficient R is of sinusoidal shape for monthly statistics with two maxima for both S3A and S3B in the NH. In the SH, the correlation coefficient varies strongly along the year. A clear peak (0.8–0.9) for both S3A and S3B is observed in June–October. The rms in the NH is within 0.25–0.32 for both S3A and S3B, with a minimum in October–January and a maximum in March–May. In the SH, rms for S3B is 0.15–0.2 in December–May and 0.09–0.14 in the other months.

Bias varies from 0.06 to 0.14 in monthly statistics in the NH. In the SH, bias is lower; it varies from 0.01 to 0.08 in monthly statistics. For S3B, bias is 0.01–0.35 lower than for S3A in all months, except April.

The fraction of matchups in the EE reflects the difference between the NH and SH and between S3A and S3B well. EE is, in general, higher for S3B with the offset up to 15 % in the NH.

As a short summary, syAOD550 validation results are slightly better for S3B; the retrieval algorithm produces better results in the SH. Obtained validation results confirm that backscatter contribution to the radiance measured at the top of the atmosphere is less critical in the SH.

There are noticeable regional differences in the performance of the retrieval algorithm, which depend on e.g. AOD load and AOD types (composition and optical properties), as well as on the properties of underlying surfaces. Retrieval quality (accuracy, precision, and coverage) varies considerably as a function of these conditions, as well as whether a retrieval is performed over land or over ocean.

Following Sogacheva et al. (2020), we intercompare validation results over 15 regions (as defined in Fig. 6) that seem likely to represent a sufficient variety of aerosol and surface conditions. These and include 11 land regions, two ocean regions, and one heavily mixed region. The land regions represent Europe (denoted by Eur), boreal (Bor), northern, eastern, and western Asia (AsN, AsE, and AsW, respectively), Australia (Aus), northern and southern Africa (AfN and AfS), South America (SA), and eastern and western North America (NAE and NAW). Southeastern China (ChinaSE), which is part of the AsE, is considered separately. The Atlantic Ocean is represented as two ocean regions, one characterised by Saharan dust outflow over the central Atlantic (AOd) and a second that includes burning outflow over the southern Atlantic (AOb). The mixed region over Indonesia (Ind) includes both land and ocean. For exact locations, see Table S2 in the Supplement.

High diversity in the validation results was observed between the selected regions (Fig. 7; Table S2 in the Supplement). The highest correlation (0.94) was found in AOb region (the number of matchups is low in this region at 22). For ChinaSE, AsN, AsE, AOd, Aus, and NAE, the correlation coefficient R was in the range 0.6–0.8, which was higher than that for the globe. For Eur and Ind, R < 0.4. For the above-mentioned regions, bias between binned syAOD and aAOD does not change much. Bias is positive in Asia, Bor, and SA regions for aAOD ≲ 1.2; bias calculated with linear regression was higher for those regions. The number of syAOD outliers, defined as |syAOD - ​​​​​​​aAOD| > 0.5, varied among the regions. In Eur, positive syAOD outliers were observed for aAOD < 0.3. For Asian and Bor regions, syAOD outliers were observed mostly for aAOD in the range of [0.2, 1.2]. More negative syAOD outliers were observed in the NAW region.

Among the land regions, the fraction of the pixels in EE was highest in Aus (81.6 %) and lowest in Bor and SA (< 30 %); for other land regions the fraction of the pixels in EE was in the 30 %–60 % interval. Over ocean, in AOb and AOd areas, the fraction of the pixels in EE was high (67.8 % and 95.5 %, respectively).

The fraction of syAOD pixels which satisfy GCOS requirements was low (< 31 %) for all regions, except for Aus (54.5 %) and AOb (68.2 %), where matchups cover low-AOD (< 0.3) conditions only.

Regional differences between syAOD and aAOD for all aerosol types including background (aAOD ≤ 0.2) AOD as well as fine-dominated and coarse-dominated AOD for selected aAOD bins are shown in Fig. 8. For most of the regions, a general tendency towards positive SY_2 AOD offsets is observed under the background conditions. Offsets are higher (up to 0.15) in Ind and SA and lower (< 0.04) in AfN, AfS, and AOd. The behaviour of the fine-dominated offset is similar for most of the regions (ChinaSE, AfN, AfS, Ind) with a gradual increase in the aAOD range of ca. 0.7–1.1. The coarse-dominated offset over Eur is underestimated by up to 0.18 for aAOD of 0.6–0.8. Over China, the coarse-dominated offset is slightly overestimated at aAOD < 0.7 and underestimated at aAOD > 1. Over bright surface with a contribution of dust aerosols (AfN), all groups show good agreement with aAOD for aAOD < 0.7. For aAOD > 0.7, syAOD for coarse-contaminated matchups is considerably underestimated. Similar offsets are observed in the NAE region, where 70 %–90 % of matchups are characterised by fine-dominated aerosols. In the possible biomass burning region (AfS), an underestimation of syAOD for coarse-dominated matchups gradually increases for aAOD > 0.3, reaching -0.9 at aAOD close to 1. Over Ind, dAOD is positive for aAOD < 0.5. Over ocean, with possible contamination of Saharan dust (AOd), offsets are constantly positive (up to 0.1) for all groups at aAOD < 1.

The syAOD offset analysis was performed for matchups which did not satisfy the GCOS requirements of |syAOD - aAOD| < 0.03 or |syAOD - aAOD| < 0.1 × aAOD (GCOS, 2016).

The syAOD relative offset, or dAOD,rel, was defined as in Eq. (1). 1dAODrel=syAOD-aAODaAOD

In Fig. 9 we show a density scatter plot for the latitude dependence of the relative offset of the syAOD for all, dual, singleN, and singleO groups of pixels for S3A. Colour indicates the fraction of the points with corresponding dAOD,rel from the total number of points within the 10∘ latitude bin. As an example, for the latitude 20–30∘ S, dAOD,rel was between -0.5 and -1 for ∼ 38 % of matchups. The magenta line shows the number of matchups in the x-axis bin.

In the NH, dAODrel was mostly positive (syAOD was higher than aAOD). In the SH, dAOD,rel is mostly positive at 30–60∘ S and mostly negative at 10–30∘ S, except for the singleN group, for which dAOD,rel is mostly positive. In both NH and SH, dAOD,rel increases towards the poles. This increase is more pronounced for the singleO group of pixels but also visible in the dual group.

The directional surface reflectance (SR) retrieved with the SYNERGY algorithm is provided in the SY_2_AOD product.

In Fig. 10 we show a density scatter plot for the dependence of the relative offset of the AOD on the retrieved SR for the dual, singleN, and singleO groups of matchups. Colour indicates the fraction of the points with corresponding dAOD,rel from the total number of points within the surface reflectance bin.

For all matchups (not shown here), as well as for the dual group (globally, as well as over the NH and SH), footprints for the dAODrel dependence on the SR are similar. For SR < 0.05 and SR > 0.35, dAOD,rel indicates that syAOD is mostly overestimated. In specified ranges, dAOD,rel increases towards outer edges. For the SR in the range of 0.05–0.35, syAOD is mostly underestimated. Underestimation is more pronounced when syAOD is retrieved with the dual processor. For the singleO group, syAOD is mostly overestimated in all SR ranges.

In Fig. 11 we show the dependence of the syAOD relative offsets on the OLCI geometry (relative azimuth (Raz), satellite zenith angle (SatZA), and sun (or solar) zenith angle (SunZA) provided in the SY_2_AOD product (North and Heckel, 2019) for the NH and SH. Colour indicates the fraction of the points with a corresponding dAOD,rel interval in the Raz, SatZA, or SunZA bins.

In the NH, positive dAOD,rel increases with Raz increasing from 50 to 80∘ and decreases with Raz increasing from 100 to 140∘. In the SH, we see a similar dependence of dAOD,rel for Raz at 50–80∘. For Raz > 90∘, positive dAOD,rel increases with a Raz increase from 150 to 180∘; a negative dAOD,rel of [-1, -0.5] is observed more often than positive [0, 0.5] dAOD,rel.

No significant dependence of dAOD,rel on the SatZA was observed. However, a greater number of negative dAOD,rel values is clearly seen in the SH.

In the NH, dAODrel is slightly positive (0–0.5) in all ranges of SunZA, except for the most extreme values. For SunZA > 80∘, the percentage of higher positive dAOD,rel (0.5–1) increases, while for SunZA < 30∘ the percentage of higher negative dAOD,rel rises. In the SH, a similar dependence was observed, except for SunZA in the range of 50–65∘, where dAOD,rel is mainly negative.

Linear fitting for combinations of syAOD550 and aAOD550 collocations has been performed with a consideration of the syAOD550 and aAOD550 uncertainties (https://se.mathworks.com/help/stats/linearmodel.predict.html, last access: 8 March 2022). For syAOD550, pixel-level uncertainties are provided in the SY_2_AOD product. For aAOD550, uncertainty of 0.01 has been considered (Eck et al., 1999). For both S3A and S3B, for all groups of matchups, bias and slope for the linear regression fits applied to the whole AOD range were improved when the syAOD and aAOD uncertainties were considered. Bias was lowered by roughly 50 %. Slope was improved by 10 %–15 %. Improvements were smaller for the singleO group of matchups (retrievals over ocean), for which the syAOD uncertainties are smallest (Sect. 6.2).

For more details, see Fig. S8 and Table S3, which are both in the Supplement.

Scatter plots for SY_2 AOD440, AOD670, AOD865, and AOD1600 are shown in Fig. 12. Clear tendencies in validation statistics were observed when comparing validation results from shorter (440 nm) to longer (1600 nm) wavelengths. Though the correlation coefficient decreases (0.65, 0.55, 0.50, and 0.40 for 440, 670, 865, and 1600 nm, respectively), the offset (0.15, 0.1, 0.07, 0.05) and rms (0.33, 0.23, 0.18, 0.16) also decrease. Note that AOD decreases significantly (except for dust aerosols) as wavelength increases.

Validation statistics for all wavelengths are slightly worse for the NH than global validation statistics (Table S4, Supplement); validation statistics for the SH are considerably better than for the NH (except for R for 1600 nm wavelength).

The syAOD440 is overestimated for all aerosol types (Fig. 13). The syAOD670 for fine-dominated matchups is in good agreement with aAOD670 for aAOD670 < 1. A similar tendency, though for narrower aAOD ranges (aAOD870 < 0.5 and aAOD1600 < 0.3), is observed for syAOD865 and syAOD1600. For all wavelengths, coarse-dominated syAOD is retrieved accurately for aAOD below ca. 0.4; above 0.4 syAOD is underestimated, and the offset between syAOD and aAOD increases with increasing aAOD.

The goodness of the predicted uncertainties was estimated with the χ2 test, as in Eq. (3) 3χ2=1N-1∑i=1Nδ‾i, with individual weighted deviation δ‾i described in Eq. (4). 4δ‾i=syAODi-aAODi-mean(syAOD-aAOD)2PUi2+AU2 If χ2 ∼ 1, prognostic uncertainties describe the AOD error well. If χ2 ≫ 1, PUs are strongly underestimated; if χ2 ≪ 1, PUs are strongly overestimated. χ2 was calculated for the whole dataset and for different AOD bins to reveal if the goodness of the PU uncertainties is AOD-dependent.

For the whole dataset, χ2 = 3.1, which means that PUs are slightly underestimated. For the binned AOD, χ2 varies strongly (Fig. 14). For aAOD < 0.4, which is ca. 90 % of all values, χ2 fits into the interval [1.8, 3.2]. Thus, for most of the matchups, PU is only slightly underestimated. For AOD > 0.4 PU underestimation is more pronounced.

No significant dependence of δ‾i on AOD error or surface reflectance provided in the SY_2_AOD product has been revealed (Fig. S9, Supplement).

Though the number of matchups in the whole dataset is high (which provides confidence in χ2 test results), it was noticed that high δ‾i (up to 155) exists, which may bias the evaluation of the PU with χ2. To remove possible contribution of the outliers to the χ2 test results, cases with δ‾i>10 (which make up less than 5 % of the total number of matchups) were removed from the analysis.

For the dataset with the removed outliers, χ2 = 1.2, which means that PUs describe the AOD error well.

The influence of δ‾i outliers is more pronounced for AOD bins, in which number of matchups per bin is lower and thus the contribution of the outliers to the results is more expected. If δ‾i outliers are removed from the binned analysis, χ2 fits the range [1, 1.45] for AOD < 0.4 (Fig. 14).

To qualitatively illustrate the accuracy of prognostic uncertainties, we show in Fig. 15 the comparison between the PU, AOD error distribution, and theoretical Gaussian distribution (with a mean of 0 and standard deviation of the syAODerror). PU distribution shows a double peak (the first peak is at ca. 0.02–0.04 for all groups; the second peak is in a range of 0.12–0.18 for different groups). For singleN, two peaks are located close to each other. Mean PU for the dual group is higher; SD is higher for the singleN group. AOD error distributions are Gauss-like with some asymmetry in the positive AOD error direction.

ED is calculated for each pixel by combining PU and AERONET uncertainties, as in Eq. (2).

For a quantitative validation, we follow (with some modifications) a new approach developed by ESA Aerosol CCI (https://climate.esa.int/media/documents/Aerosol_cci_PVIR_v1.2_final.pdf, last access: 25 February 2022). A synthetic cumulative distribution of ED is calculated assuming a Gaussian error distribution (normalised to a total integral of 1) with standard deviation of ED. In the next step, this synthetic error frequency distribution is compared with the AOD error. We calculate and subtract the mean bias from the AOD error distribution to make it more symmetric for direct comparison to the synthetic distribution (which by definition is always symmetric). Bias correction results for S3A all, dual, and singleN (0.07, 0.04, and 0.12, respectively) are shown in Fig. 16.

Finally, we calculate an average correction factor for the synthetic distribution (and thus the prognostic uncertainties) in relation to the mean-bias-corrected error distributions as the ratio of the absolute means of both distributions. Correction factors are different for all matchups and for the dual and singleN groups. A small correction is needed for all and singleN (0.80 and 1.1, respectively). For the dual group, the correction is stronger (0.67); ED should be lowered.

However, the correction method applied here does not equally improve ED in all ranges. The correction factor is biased by the number of pixels with small (< 0.2) absAODerror. Thus, for those cases the correction works well; overestimated ED is lowered by 0.8 and 0.65 for the all and dual groups. For absAODerror ≳ ​​​​​0.3, where ED is underestimated, correction degrades ED and increases disagreement between ED and AOD error. A possible solution can be to perform correction separately for different absAODerror ranges, but setting specific relations for different groups between ED and absAODerror makes the analysis very complicated.

Sayer et al. (2020) suggested the analysis of the potential of the PU to discriminate between (“good” and “bad”) pixels with likely small or large errors. Instead of PU, we perform analysis of the ED, which, besides PU, includes uncertainties of the ground-based measurements.

To estimate the potential of ED, we plot the absolute errors, which 38 % of all pixels are below, as a function of binned ED (Fig. 17). We then repeat this for the fractions 68 % and 95 %. These percentages relate to 0.5σ, 1σ, and 2σ (where σ is a standard width) for normal error distributions in each bin (along the vertical axis). Theoretically, expected values are shown as dashed lines in black, red, and blue. The number of pixels per ED bin is shown as a grey dashed line.

The percentile plots show reasonable agreement (within statistical noise) with the theoretical lines of 38 % and 68 % for the majority of the validation points in the lower range of ED (up to 0.05–0.2) for all groups, with underestimation of the true error at higher values of ED for the 38 % and 68 % lines. For the dual-view case, ED overestimates the true error, while for the single-view case the true error is higher than the ED prediction, especially at higher values of ED (ED ≳ 0.2).